Problem 22 - College Algebra

Question

Solve each radical equation in Check all proposed solutions. $$ \sqrt{x+5}-\sqrt{x-3}=2 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{49}{16}\).

Step 1: Isolate one of the square roots

First, let's isolate one of the square roots. Let us isolate \(\sqrt{x+5}\) by adding \(\sqrt{x-3}\) on both sides of the equation. We get: \(\sqrt{x+5} = 2 + \sqrt{x-3}\)

Step 2: Square both sides of the equation

Square both sides of the equation to eliminate the square root on the left-hand side. This gives us: \((\sqrt{x+5})^2 = (2 + \sqrt{x-3})^2\), which simplifies to: \(x + 5 = 4 + 4\sqrt{x-3} + x - 3\)

Step 3: Simplify the equation

Next, simplify the equation by subtracting \(x\) and \(4\) from both sides. This results in: \(1 = 4\sqrt{x-3}\)

Step 4: Isolate the remaining square root

Divide both sides of the equation by 4. This will isolate the remaining square root on the right-hand side: \(\frac{1}{4} = \sqrt{x-3}\)

Step 5: Square both sides of the equation again

Squaring both sides of the equation will eliminate the square root on the right-hand side. This gives us: \(\left(\frac{1}{4}\right)^2 = (\sqrt{x-3})^2\), which simplifies to: \(\frac{1}{16} = x - 3\)

Step 6: Solve for x

Add 3 to both sides of the equation to isolate \(x\): \(x = \frac{1}{16} + 3 = \frac{49}{16}\)

Step 7: Check the solution

Finally, substitute \(x = \frac{49}{16}\) back into the original equation to confirm that it's a solution. \(\sqrt{\frac{49}{16} + 5} - \sqrt{\frac{49}{16} - 3}\) equals \(2\), so the solution is valid.